Chapter 42, Problem 1OQ

(i)

To determine

The principal quantum number of the initial state of an atom as it emits an ${\text{M}}_{\beta }$ line in an x-ray spectrum.

Answer to Problem 1OQ

Option (e), 5 is the principal quantum number of the initial state of an atom as it emits an ${\text{M}}_{\beta }$ line in an x-ray spectrum.

Explanation of Solution

$M_{\beta }$ line in an x-ray spectrum shows that electrons falls into the $\text{M}$ shell, and the $\beta$ shows the difference between the shells. All states having the same principal quantum number are said to form a shell. Shells are identified by K, L, M….which designate the states for which $n=1,2,3,\mathrm{...}$.

The principal quantum number corresponding to the M shell is 3, and the $\beta$ shows that the electron comes from two shells above M, considering the order of shells, the electrons will fall from the $\text{O}$ shell to the M shell for which $n=5$.

Conclusion:

Since the electrons emits from the $\text{O}$ shell to the M shell, option (e) is correct.

Since the electrons fails to emit from the K shell, option (a) is incorrect.

Since the electrons are not coming from the L shell, option (b) is incorrect.

Since the electrons cannot emit from M shell to produce ${\text{M}}_{\beta }$ line, option (c) is incorrect.

Since the electrons are not excited from the N shell, option (d) is incorrect.

(ii)

To determine

The principal quantum number of the final state of an atom as it emits an ${\text{M}}_{\beta }$ line in an x-ray spectrum.

Answer to Problem 1OQ

Option (c), 3 is the principal quantum number.

Explanation of Solution

$M_{\beta }$ line in an x-ray spectrum shows that electrons falls into the $\text{M}$ shell, and the $\beta$ shows the difference between the shells.

The principal quantum number corresponding to the M shell is 3, and the $\beta$ shows that the electron emits from two shells above M, considering the order of shells, the electrons will fall from the $\text{O}$ shell to the M shell, correspondingly the principal quantum number of the transition will be from $n=5$ to $n=3$. Thus, the principal quantum number of the final state will be $3$.

Conclusion:

Since final state is M shell whose principal quantum number is $3$, option (c) is correct.

The principal quantum number of the final state is not $2$, Option (a) is incorrect.

The principal quantum number of the final state is not $4$, option (d) is incorrect.

The principal quantum number of the final state is not $5$, option (e) is incorrect.